Use the image to answer the question.

Question

A bird’s eye view of a pool table is shown.

Mark is playing pool. On a pool table there are 6 holes that you have to hit the balls into. Four of the holes are located at the four corners of the table, and the other two holes are located at the midpoints of the long sides of the table. These two holes are indicated on the image at points F and C, and a line segment has been drawn connecting these two points. Line segment GH¯¯¯¯¯¯¯¯
has been drawn as the perpendicular bisector of FC¯¯¯¯¯¯¯¯
. GH¯¯¯¯¯¯¯¯
intersects FC¯¯¯¯¯¯¯¯
at point J. Mark only has one ball left to hit in, the 8-ball. The 8-ball is located at point I, which is on perpendicular bisector GH¯¯¯¯¯¯¯¯
. Mark is trying to decide if the 8-ball is closer to the hole at point F or the hole at point C.

a. Use your knowledge about perpendicular bisectors to prove to Mark that the 8-ball at point I is equidistant from the hole at point F and the hole at point C. You can choose if you want to write a paragraph proof or a two-column proof, but be sure to include a reason for each of your statements.

b. Name another point in the diagram that would also be equidistant from points F and C. Explain how you know.



(4 points)

Short Answer Rubric (4 points)
Points Awarded Criteria
4
The answer is very clear.

The answer is fully supported by details.

There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

3
The answer is mostly clear.

The answer is supported by mostly relevant details.

There are a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.

2
The answer is somewhat clear.

Details are weak or not related.

There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.

1
The answer is not clear.

There are few or no supporting details.

There are many errors in grammar, usage, and mechanics. Errors interfere with meaning.

0 The question is not answered.
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Answer 1

Part A: Proof that point I is equidistant from points F and C

Proof:

1. Statement: Line segment GH is the perpendicular bisector of line segment FC.

   • Reason: By definition of a perpendicular bisector, it is a line that is perpendicular to a segment at its midpoint.

2. Statement: Point J is the midpoint of segment FC.

   • Reason: Since GH is the perpendicular bisector of FC, it intersects FC at its midpoint.

3. Statement: Point I is located on the perpendicular bisector GH.

   • Reason: It is given in the problem that the 8-ball is at point I, which lies on line segment GH.

4. Statement: According to the properties of perpendicular bisectors, any point on the perpendicular bisector of a segment is equidistant from the endpoints of that segment.

   • Reason: This is a geometric property of perpendicular bisectors.

5. Conclusion: Since point I lies on the perpendicular bisector GH of segment FC, it follows that the distances from point I to points F and C are equal.

   • Final Statement: Therefore, the 8-ball at point I is equidistant from both holes at points F and C.

Part B: Another Point Equidistant from F and C

Another point in the diagram that would also be equidistant from points F and C is point J, which is the intersection point of line segment GH (the perpendicular bisector) with line segment FC.

Explanation: Since point J is the midpoint of segment FC (as established in part A), and by the properties of midpoints and perpendicular bisectors, J is equidistant from points F and C. Thus, the distances from point J to points F and C are equal, just like point I, confirming that J is also equidistant from points F and C.